In this problem, the black triangle has undergone a translation to form the red triangle, which is reflected in the coordinates of the vertices. Let's analyze the translation in vector notation.

We can observe how each vertex of the black triangle moves to its corresponding vertex in the red triangle:

1. The black vertex $U$ moves to the red vertex $U'$.
2. The black vertex $V$ moves to the red vertex $V'$.
3. The black vertex $W$ moves to the red vertex $W'$.
4. The black vertex $X$ moves to the red vertex $X'$.

Let's break this down step by step:

Coordinates of Points Before Translation (Black)

• $U = (-6, -1)$
• $V = (-3, 3)$
• $W = (1, 3)$
• $X = (1, 0)$

Coordinates of Points After Translation (Red)

• $U' = (-6, 3)$
• $V' = (-3, 7)$
• $W' = (1, 7)$
• $X' = (1, 4)$

Translation Vector

The translation vector can be found by determining the difference between corresponding points. For example, for the point $U$ to $U'$:

$\text{Translation vector} = U' - U = (-6, 3) - (-6, -1) = (0, 4)$

We repeat the same for all the points, and it becomes evident that the translation vector is the same for all points:

$\text{Translation vector} = (0, 4)$

Final Answer

The correct vector notation for the given translation is $\mathbf{T} = \langle 0, 4 \rangle$, which means the figure was translated 4 units upward.

Would you like any further details or explanations?

Here are five questions related to this problem:

1. What would the translation vector be if the triangle had been translated downward instead?
2. How would the translation vector change if the figure was also shifted horizontally?
3. How can you represent a reflection transformation in vector notation?
4. What is the geometric effect of applying the vector $\langle 4, -4 \rangle$ to a figure?
5. Can you combine multiple translations into a single vector notation?

Tip: When working with translations, always check that the same vector applies to all points, ensuring that the figure remains congruent after the transformation.